A note on a deformation of Dirichlet's class number formula
For a prime \(p\), let \(\mathbb{Q}(\sqrt p)\) be the real quadratic field. Then, \textit{T. Ono} gave a deformation of the classical Dirichlet class number formula in case \(p\equiv 1\pmod 4\) [Algebraic analysis, Pap. dedic. M. Sato, Vol. 2, 659-666 (1989; Zbl 0683.12005)] and the author gave it in case \(p\equiv 3\pmod 4\) [On a deformation of Dirichlet's class number formula, Res. Bull. Tokushima Bunri Univ. 42]. In this paper, for any real quadratic field \(\mathbb{Q}(\sqrt m)\) the author gives by using the same method the following deformation: for the class number \(h\), the fundamental unit \(\varepsilon\) and the discriminant \(d\) of \(\mathbb{Q}(\sqrt m)\), we put \(w=0\) if \(d\) has at least two distinct prime factors and \(w=1\) otherwise. Then \[ \sqrt{m^ w}\varepsilon^ h=2\bigl(\sum^{v-1}_{j=0}d_ j\bigr)+d_ v\quad(v=\varphi(d)/4), \] where \(d_ j\) are determined by the following recurrence relation: \(jd_ j=\sum^ j_{t=1}\alpha_ td_{j-t}\) \((d_ 0=1,\;1\leq j\leq v)\), \(\alpha_ t=-(\varphi(d)/\varphi(d/n)\mu(d/n)-\chi(t)\sqrt d)/2\), \(n=(t,d)\), \((1\leq t\leq v)\) \((\varphi\) denotes Euler's function, \(\chi\) a Kronecker character belonging to \(\mathbb{Q}(\sqrt m))\).
scientific article; zbMATH DE number 97929